Two-dimensional (2D) Cairo pentagon materials have attracted tremendous interest due to their exotic physical properties. Based on first-principles calculations, we predict that monolayer penta-${\mathrm{BP}}_{2}$ is a promising Cairo pentagon-based 2D material simultaneously harboring a negative Poisson ratio and intrinsic superconductivity. The Poisson ratio exhibits strong anisotropy with alternating positive-negative signs, and its value along the diagonal direction is more negative than that of pentagraphene. The medium-strength coupling between the Fermi-surface electrons and a phonon mode majorly originating from the ${\mathrm{P}}_{2}$ dimers leads to a superconducting transition estimated to occur at ${T}_{\mathrm{c}}\ensuremath{\sim}7.7\phantom{\rule{0.28em}{0ex}}\mathrm{K}$. Moreover, we find that fluorination/chlorination can efficiently engineer the superconductivity and mechanical properties of penta-${\mathrm{BP}}_{2}$: achieving higher ${T}_{\mathrm{c}}$ of 9.6/10.5 K, turning Poisson's ratio from negative to positive and enhancing Young's modulus. These results render penta-${\mathrm{BP}}_{2}$ an appealing platform to investigate unique superconductivity and mechanical properties in the Cairo pentagon lattice.